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Properties

Label 2-21e2-63.20-c1-0-27
Degree $2$
Conductor $441$
Sign $0.371 - 0.928i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.23 + 1.28i)2-s + (1.71 + 0.266i)3-s + (2.32 + 4.02i)4-s + (−1.16 − 2.01i)5-s + (3.47 + 2.80i)6-s + 6.82i·8-s + (2.85 + 0.911i)9-s − 6.01i·10-s + (−3.78 − 2.18i)11-s + (2.90 + 7.50i)12-s + (1.14 − 0.660i)13-s + (−1.45 − 3.76i)15-s + (−4.15 + 7.18i)16-s − 5.78·17-s + (5.20 + 5.71i)18-s − 0.675i·19-s + ⋯
L(s)  = 1  + (1.57 + 0.911i)2-s + (0.988 + 0.153i)3-s + (1.16 + 2.01i)4-s + (−0.521 − 0.903i)5-s + (1.41 + 1.14i)6-s + 2.41i·8-s + (0.952 + 0.303i)9-s − 1.90i·10-s + (−1.14 − 0.658i)11-s + (0.838 + 2.16i)12-s + (0.317 − 0.183i)13-s + (−0.376 − 0.972i)15-s + (−1.03 + 1.79i)16-s − 1.40·17-s + (1.22 + 1.34i)18-s − 0.154i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.371 - 0.928i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.371 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.18992 + 2.15831i\)
\(L(\frac12)\) \(\approx\) \(3.18992 + 2.15831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 - 0.266i)T \)
7 \( 1 \)
good2 \( 1 + (-2.23 - 1.28i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.16 + 2.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.78 + 2.18i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.14 + 0.660i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.78T + 17T^{2} \)
19 \( 1 + 0.675iT - 19T^{2} \)
23 \( 1 + (4.81 - 2.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.86 - 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.47 - 2.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 + (-3.29 - 5.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.89 + 6.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.246 - 0.427i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.14iT - 53T^{2} \)
59 \( 1 + (2.15 + 3.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.77 + 1.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 - 4.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.17iT - 71T^{2} \)
73 \( 1 - 15.1iT - 73T^{2} \)
79 \( 1 + (-5.30 + 9.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.32 + 9.22i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.32T + 89T^{2} \)
97 \( 1 + (-12.7 - 7.36i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.66353822258639046288772936853, −10.58116693908649461845947940709, −9.008118292971408494587725972087, −8.220452093258340700634912146691, −7.67062509778036507634305538158, −6.51809875995586521637853206452, −5.31423013443525838598384973934, −4.50983785587852002190693368316, −3.66701077298779721904415341640, −2.52629410631714140916981738547, 2.13981437140464848746113273658, 2.79031522237495488870963649703, 3.92514696595036372624531276809, 4.62324497434682596802413878721, 6.13278757478620143028634313455, 7.05519954770598407545175253009, 8.028066719975965130958870628242, 9.462723683507157822845808881379, 10.52519198864677656345175759683, 10.95491933858821614447777153502

Graph of the $Z$-function along the critical line